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Weekly Handouts
Prerequisites to Group Theory Notation: ' \exists means "there exists". \forall means "for all". \in means "in". X\times Y is the set of all pairs (x,y) with x\in X, y\in Y . This is called the '''Cartesian Product '''of X and Y . For a much more complete list, see this. '''Sets. ' First of all, you should know what a set is. From Wikipedia, "a set is a collection of distinct object". I give a few examples: \mathbb{Z} is the set of all integers \{...-2,-1,0,1,2...\} \mathbb{N} is the set of all positive integers \{1,2,3...\} \mathbb{Q} is the set of all rational numbers, i.e. numbers that can be written in the form \dfrac{a}{b} for a,b integers. \mathbb{R} is the set of all real numbers. \mathbb{C} is the set of all complex numbers, i.e. numbers of the form a+bi where a,b are reals. '''Functions. Next, we look at some definitions related to functions. Whenever you see the notation f:X\rightarrow Y where X and Y are sets, that means that f can take in any input from X and the unique output must be in Y . X is called the domain and Y the codomain. However, not every value in Y has to be an output. (If every is, then we have a surjective function.) Example: ' f:\mathbb{R}\rightarrow \mathbb{R}, f(x)=x^2. Note how we can take any input from the domain \mathbb{R} and clearly every output will be a real number. However, we remark that note every in the codomain is reachable. For instance, what about solving x^2=-1? We will know look at what it means for a function to be injective, surjective, or bijective. '''Definitions: ' '(a) ' f:X\rightarrow Y is injective if f(a)=f(b)\Rightarrow a=b where a,b are in X . '(b) ' f:X\rightarrow Y is surjective if for every y\in Y there exist an x in X such that f(x)=y. '© ' f:X\rightarrow Y is bijective if it is both injective and surjective. Try to internalize these definitions. They are quite useful. '''Exercises: Determine whether the following are injective, surjective, both (bijective) or neither. 1. ' f:\mathbb{R}\rightarrow \mathbb{R}, f(x)=x^3. '''2. ' f:\mathbb{R}\rightarrow \mathbb{R}, f(x)=\dfrac{1}{1+x^2}. '''3. f:[0,\infty) \rightarrow \mathbb{R}, f(x)=\sqrt{x}. Group Theory: Introduction Now we are ready to begin a discussion of group theory. Naturally, we first define a group. A group (G,*) consists of a set G and a binary operation * that satisfies the following properties: (a) 'If g_1 and g_2 are \in G , then g_1*g_2\in G . '(b) ' \forall a,b,c \in G, a*(b*c)=(a*b)*c '© ' \exists i\in G such that i*g=g*i=g \forall g\in G. This element i is denoted by e '(d) ' \forall g\in G, \exists h \in G such that g*h=h*g=e . We will denote h by g^{-1} . Note, do not confuse * with multiplication. * can be many other things like addition or even composition of functions (which we will discuss more later). Also, note that though we have condition '(b), we don't have anything saying that a*b=b*a . Though this is not always true, if this happens to be true for a specific group we call that group Abelian. Exercises: ''' '''1. '''Prove that a group has only one identity element. '''2. '''Prove that every element in a group has only one inverse. '''Exercises: Determine which of the following are groups. If a group, determine if Abelian. 1. ' (\mathbb{Z},+) '''2. ' (\mathbb{C},\times) where \times is multiplication. '3. ' (\mathbb{N},+) '''4. '''The set of all 2\times 2 matrices with operation matrix addition. '''5. '''The set of all 2\times 2 matrices with operation matrix multiplication. '''Some more complicated examples of Groups: First, we look at permutations. Consider the set S=\{1,2,3...n\} . Then, a permutation of this set is a bijection from ' ' S\rightarrow S . For example, consider S=\{1,2,3,4\} . A possible permutation would be the bijection S\rightarrow S, \{1,2,3,4\}\rightarrow \{2,4,1,3\}. We then define S_n to be the set of all possible permutations of \{1,2...n\} . And if we take the operation to be composition of permutations denoted by \circ , then (S_n,\circ) is a group. Another similar example is the following. If you take the set of invertible functions f:\mathbb{C}\rightarrow \mathbb{C} then if you take the operation of function composition, we have a group. NOTE: '''Often, when the operation is understood, we will abreviate a group (G,*) by simply G . Other times, we will write G_{*} . Now we will introduce some very important terminology for Group Theory. '''Subgroups. This is rather straightforward. (H,*) is a subgroup of (G,*) if H is a subset of G . Note how they have the same operation. Notationally, this is written (H,*)\subseteq (G,*) . A simple example is \mathbb{Z}_{+}\subseteq \mathbb{R}_{+} . Cosets. For a group G , and element g\in G , and a subgroup H , we define the left coset of H in G to be \{gh| h\in H\} and the right coset to be \{hg| h\in H\} . Cardinality. If a group has a finite number of elements, then its cardinality is the number of elements in it. Symmetry Groups This topic will be very important later for things such as applications of Burnside's Lemma which we will discuss later. Because of this, it is critical to understand this well. First of all, recall the discussion of S_n. As mentioned before, any rearrangement of the numbers 1,2....n is called a permutation. The rearrangement that leaves it the same is the identity permutation. We will write a permutation as \left( \begin{array}{ccccc} 1 & 2 & 3 & ... & n\\ - & - & - & ... & - \end{array} \right) where the row on the bottom is the rearrangement of the top row. To give a concrete example, \left( \begin{array}{ccc} 1 & 2 & 3\\ 2 & 1 & 3 \end{array} \right) symbolizes the rearrangement \{2,1,3\} of \{1,2,3\}. We will denote a permutation by P or other letter. Next, I will define "products" of permutations. Consider the permutations P= \left( \begin{array}{cccc} 1 & 2 & 3 & 4\\ 2 & 1 & 4 & 3\end{array} \right) and Q= \left( \begin{array}{cccc} 1 & 2 & 3 & 4\\ 1 & 4 & 3 & 2\end{array} \right). Then, how do you find the "product" PQ of P and Q ? To do this note that in P , 1 goes to 2, 2 to 1, 3 to 4, 4 to 3. In Q , 1 goes to 2, 2 to 4, 3 to 3, and 4 to 2. Therefore, with the product, 1 goes to 4, 2 to 1, 3 to 2, and 4 to 3. Therefore, PQ= \left( \begin{array}{cccc} 1 & 2 & 3 & 4\\ 4 & 1 & 2 & 3\end{array} \right). Exercise 1. If P= \left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 2 & 1 & 5 & 3 & 4\end{array} \right) and Q= \left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 5 & 4 & 4 & 3 & 1\end{array} \right). find PQ . Now, we will call the identity permutation I . You should convince yourself that for any permutation P there is another permutation P^{-1} s.t. PP^{-1}=I . Using this, we see how the set of all such permutations ( S_n ) is a group. Geometric Interpretation of Symmetry Groups We want a way to find the symmetries of this polygon and write them in a notation. For polygons, the two types of symmetry are reflexive and rotational. Reflexive symmetry is when there is a line such that when the polygon is flipped (aka reflected) across this line it is unchanged. As an example, imagine an equilateral triangle. Then, draw a line through a vertex and the midpoint of the base. This line will be a line of reflexive symmetry since flipping the triangle across this line keeps it the same. Similarly, rotational symmetry is when we can rotate the polygon an angle and it stays the same. In fact, we can represent these symmetries with the matrices used above. I will now demonstrate how. Consider a regular pentagon with its vertices labelled 1 through 5 clockwise. Then, consider the line that goes through 1 and the midpoint of the line segment between 3 and 4 . If we reflect about this line, 1 goes to 1, 2 to 5, 3 to 4, 4 to 5, and 5 to 2. We can write this as symmetry as P= \left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 1 & 5 & 4 & 3 & 2\end{array} \right). Now, consider the rotational symmetry where we rotate the pentagon 72^{\circ} clockwise. Then we have the symmetry P= \left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1\end{array} \right). Exercises 1. A regular 15-gon has L lines of symmetry, and the smallest positive angle for which it has rotational symmetry is R degrees. What is L+R ? (A) 24; (B) 27; © 32; (D) 39; (E) 54 (2015 AMC 12, Problem 7) Note: You don't need to use any of the matrix stuff. This just tests your understanding of reflectional and rotational symmetry. 2. Write down (in matrix form) the symmetries of a (a) square, (b)regular hexagon, © 45-45-90 right triangle. Now we are ready to move on to some more theoretical topics in Group Theory. Dihedral Groups. A dihedral group with 2n elements consists the n rotational and reflectional symmetries of a regular n- gon. We will denote this group by D_{2n} . '''Exercise: '''If we let a rotation of 2\pi/n clockwise be denoted by R and F denotes a flip with respect to a fixed axis, do you see why F^2=1, F^3=F ... ? Also, why does D_{2n}=\{1,R...R^{n-1},F,FR....FR^{n-1}\} ? Note: we are implicitely using the operation described above (with the matrices). Group Actions Group actions are essentially a generalization of the symmetry concept just studied. Recall the definition of a '''Cartesian Product. '''Now we are ready to define a group action. '''Definition: '''Consider a group (G,*) and a set X . A group action is a binary operation \cdot: G\times X\rightarrow X; (g,x)\rightarrow g\cdot x such that 1. e\cdot x=x\ \forall x\in X 2. g\cdot (h\cdot x)=(g*h)\cdot x\ \forall g,h\in G, x\in X . We say " G acts on X ". Orbits and Stabilizers .... and finally Burnside's Lemma! See an article I will send out/post. This will give a better explanation.